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21: DLMF Project News

error generating summary

22: 31.10 Integral Equations and Representations

§31.10 Integral Equations and Representations

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Kernel Functions

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Kernel Functions

… ►For integral equations for special confluent Heun functions (§31.12) see Kazakov and Slavyanov (1996).

23: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►Solutions are called roots of the equation, or zeros of $f$. … ►and the solutions are called fixed points of $\varphi$. … ►For fixed-point methods for computing zeros of special functions, see Segura (2002), Gil and Segura (2003), and Gil et al. (2007a, Chapter 7). … ►Corresponding numerical factors in this example for other zeros and other values of $j$ are obtained in Gautschi (1984, §4). …

24: 10.2 Definitions

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§10.2(ii) Standard Solutions

… ►This solution of (10.2.1) is an analytic function of $z\in ℂ$, except for a branch point at $z=0$ when $\nu$ is not an integer. … ►Each solution has a branch point at $z=0$ for all $\nu \in ℂ$. … ►The notation ${𝒞}_{\nu }\left(z\right)$ denotes $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. ►

§10.2(iii) Numerically Satisfactory Pairs of Solutions

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25: 29.6 Fourier Series

§29.6 Fourier Series

… ►When $\nu \ne 2n$, where $n$ is a nonnegative integer, it follows from §2.9(i) that for any value of $H$ the system (29.6.4)–(29.6.6) has a unique recessive solution $A_0,A_2,A_4,\mathrm{\dots }$; furthermore …In addition, if $H$ satisfies (29.6.2), then (29.6.3) applies. ►In the special case $\nu =2n$, $m=0,1,\mathrm{\dots },n$, there is a unique nontrivial solution with the property $A_{2p}=0$, $p=n+1,n+2,\mathrm{\dots }$. This solution can be constructed from (29.6.4) by backward recursion, starting with $A_{2n+2}=0$ and an arbitrary nonzero value of $A_{2n}$, followed by normalization via (29.6.5) and (29.6.6). …

26: Daniel W. Lozier

… ►Army Engineer Research and Development Laboratory in Virginia on finite-difference solutions of differential equations associated with nuclear weapons effects. …His research interests have centered on numerical analysis, special functions, computer arithmetic, and mathematical software construction and testing. … ►He has also served several terms as an officer of the SIAM Activity Group on Orthogonal Polynomials and Special Functions. …

27: 3.6 Linear Difference Equations

… ►Many special functions satisfy second-order recurrence relations, or difference equations, of the form … ► … ► … ►A new problem arises, however, if, as $n\to \mathrm{\infty }$, the asymptotic behavior of $w_n$ is intermediate to those of two independent solutions $f_n$ and $g_n$ of the corresponding inhomogeneous equation (the complementary functions). … ►Thus the asymptotic behavior of the particular solution ${𝐄}_n\left(1\right)$ is intermediate to those of the complementary functions $J_n\left(1\right)$ and $Y_n\left(1\right)$; moreover, the conditions for Olver’s algorithm are satisfied. …

28: Bibliography O

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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations. Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.

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External Links:

Document, MathReview (J. Todd), ZentralBlatt

Referenced by:

§10.21(ii).

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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.

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External Links:

Document, MathReview (T. E. Hull), ZentralBlatt

Referenced by:

§13.19, §13.7(ii).

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Exponentially-improved asymptotic solutions of ordinary differential equations I: The confluent hypergeometric function. SIAM J. Math. Anal. 24 (3), pp. 756–767.

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External Links:

ISSN 0036-1410, Document, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§10.17(v), §13.7(iii), (9.7.18), (9.7.19), (9.7.20), (9.7.21), (9.7.22), (9.7.23), §9.7(v).

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Numerical solution of Riemann-Hilbert problems: Painlevé II. Found. Comput. Math. 11 (2), pp. 153–179.

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ISSN 1615-3375, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§32.17, §32.17.

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Solution of Equations in Euclidean and Banach Spaces. Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London.

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Notes:

Third edition of Solution of equations and systems of equations

External Links:

MathReview (J. Burlak), ZentralBlatt

Referenced by:

§3.3(v), §3.8(iii).

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29: Bibliography S

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The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

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External Links:

ISSN 1095-7170, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(v).

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Computing the complex zeros of special functions. Numer. Math. 124 (4), pp. 723–752.

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ISSN 0029-599X, Document, FullText, MathReview (Miodrag Petković), ZentralBlatt

Referenced by:

§10.21(ix), §10.21(ix), §10.74(vi), §10.74(vi), §3.8(v), §3.8(v).

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On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.

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ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

Referenced by:

§29.17(i), §29.8.

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Special Functions: A Unified Theory Based on Singularities. Oxford Mathematical Monographs, Oxford University Press, Oxford.

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External Links:

ISBN 0-19-850573-6, MathReview, ZentralBlatt

Referenced by:

Ch.2, §31.12, §31.17(ii).

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Tables of Airy Functions and Special Confluent Hypergeometric Functions. Pergamon Press, New York.

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Notes:

Translated from the Russian by D. G. Fry

External Links:

MathReview (W. F. Freiberger), ZentralBlatt

Referenced by:

(9.13.19), (9.13.20), (9.13.21), §9.13(i), 1st item.

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30: Bibliography C

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The hypergeometric function and the $R$-function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

External Links:

ISSN 0373-1243, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.27(vi).

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Incomplete Hankel and modified Bessel functions: A class of special functions for electromagnetics. IEEE Trans. Antennas and Propagation 52 (12), pp. 3373–3389.

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ISSN 0018-926X, Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§10.46.

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Special polynomials associated with rational solutions of the fifth Painlevé equation. J. Comput. Appl. Math. 178 (1-2), pp. 111–129.

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ISSN 0377-0427, Document, MathReview (Davide Batic), ZentralBlatt

Referenced by:

§32.8(v), §32.9(ii).

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Painlevé Equations—Nonlinear Special Functions: Computation and Application. In Orthogonal Polynomials and Special Functions, F. Marcellàn and W. van Assche (Eds.), Lecture Notes in Math., Vol. 1883, pp. 331–411.

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External Links:

ISBN 3-540-31062-2, MathReview (Marta Mazzocco), ZentralBlatt

Referenced by:

Ch.32.

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Algorithms for special functions. I. Numer. Math. 4, pp. 403–419.

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External Links:

ISSN 0029-599X, Document, MathReview (A. S. Householder), ZentralBlatt

Referenced by:

§5.24(ii).

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